# Properties

 Label 560.2.q.h Level 560 Weight 2 Character orbit 560.q Analytic conductor 4.472 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} + q^{15} + ( -4 + 4 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + ( 2 - 3 \zeta_{6} ) q^{21} + \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} + 9 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} + ( 1 + 2 \zeta_{6} ) q^{35} -4 \zeta_{6} q^{37} + q^{41} -9 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + ( 8 - 5 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{51} + ( 10 - 10 \zeta_{6} ) q^{53} + 2 q^{55} -2 q^{57} + ( -10 + 10 \zeta_{6} ) q^{59} -9 \zeta_{6} q^{61} + ( 2 + 4 \zeta_{6} ) q^{63} + ( 5 - 5 \zeta_{6} ) q^{67} + q^{69} -14 q^{71} + ( -12 + 12 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( 4 - 6 \zeta_{6} ) q^{77} + 14 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -11 q^{83} -4 q^{85} + ( 9 - 9 \zeta_{6} ) q^{87} + 15 \zeta_{6} q^{89} -4 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} -18 q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + q^{5} + 5q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} + q^{5} + 5q^{7} + 2q^{9} + 2q^{11} + 2q^{15} - 4q^{17} - 2q^{19} + q^{21} + q^{23} - q^{25} + 10q^{27} + 18q^{29} + 4q^{31} - 2q^{33} + 4q^{35} - 4q^{37} + 2q^{41} - 18q^{43} - 2q^{45} + 11q^{49} + 4q^{51} + 10q^{53} + 4q^{55} - 4q^{57} - 10q^{59} - 9q^{61} + 8q^{63} + 5q^{67} + 2q^{69} - 28q^{71} - 12q^{73} + q^{75} + 2q^{77} + 14q^{79} - q^{81} - 22q^{83} - 8q^{85} + 9q^{87} + 15q^{89} - 4q^{93} + 2q^{95} - 36q^{97} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 2.50000 0.866025i 0 1.00000 + 1.73205i 0
401.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 2.50000 + 0.866025i 0 1.00000 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.h 2
4.b odd 2 1 280.2.q.a 2
7.c even 3 1 inner 560.2.q.h 2
7.c even 3 1 3920.2.a.m 1
7.d odd 6 1 3920.2.a.y 1
12.b even 2 1 2520.2.bi.a 2
20.d odd 2 1 1400.2.q.e 2
20.e even 4 2 1400.2.bh.b 4
28.d even 2 1 1960.2.q.k 2
28.f even 6 1 1960.2.a.e 1
28.f even 6 1 1960.2.q.k 2
28.g odd 6 1 280.2.q.a 2
28.g odd 6 1 1960.2.a.i 1
84.n even 6 1 2520.2.bi.a 2
140.p odd 6 1 1400.2.q.e 2
140.p odd 6 1 9800.2.a.r 1
140.s even 6 1 9800.2.a.bc 1
140.w even 12 2 1400.2.bh.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 4.b odd 2 1
280.2.q.a 2 28.g odd 6 1
560.2.q.h 2 1.a even 1 1 trivial
560.2.q.h 2 7.c even 3 1 inner
1400.2.q.e 2 20.d odd 2 1
1400.2.q.e 2 140.p odd 6 1
1400.2.bh.b 4 20.e even 4 2
1400.2.bh.b 4 140.w even 12 2
1960.2.a.e 1 28.f even 6 1
1960.2.a.i 1 28.g odd 6 1
1960.2.q.k 2 28.d even 2 1
1960.2.q.k 2 28.f even 6 1
2520.2.bi.a 2 12.b even 2 1
2520.2.bi.a 2 84.n even 6 1
3920.2.a.m 1 7.c even 3 1
3920.2.a.y 1 7.d odd 6 1
9800.2.a.r 1 140.p odd 6 1
9800.2.a.bc 1 140.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 9 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4}$$
$59$ $$1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4}$$
$61$ $$1 + 9 T + 20 T^{2} + 549 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$
$71$ $$( 1 + 14 T + 71 T^{2} )^{2}$$
$73$ $$1 + 12 T + 71 T^{2} + 876 T^{3} + 5329 T^{4}$$
$79$ $$1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 11 T + 83 T^{2} )^{2}$$
$89$ $$1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 18 T + 97 T^{2} )^{2}$$